Mathematics > Classical Analysis and ODEs

Title:Nonlinear phase unwinding of functions

Abstract: We study a natural nonlinear analogue of Fourier series. Iterative Blaschke
factorization allows one to formally write any holomorphic function $F$ as a
series which successively unravels or unwinds the oscillation of the function
$$ F = a_1 B_1 + a_2 B_1 B_2 + a_3 B_1 B_2 B_3 + \dots$$ where $a_i \in
\mathbb{C}$ and $B_i$ is a Blaschke product. Numerical experiments point
towards rapid convergence of the formal series but the actual mechanism by
which this is happening has yet to be explained. We derive a family of
inequalities and use them to prove convergence for a large number of function
spaces: for example, we have convergence in $L^2$ for functions in the
Dirichlet space $\mathcal{D}$. Furthermore, we present a numerically efficient
way to expand a function without explicit calculations of the Blaschke zeroes
going back to Guido and Mary Weiss.